Captain Emily has a ship, the H.M.S Crimson Lynx. The ship is five furlongs from the dread pirate Umaima and her merciless band of thieves. If her ship hasn't already been hit, Captain Emily has probability $\dfrac{3}{5}$ of hitting the pirate ship. If her ship has been hit, Captain Emily will always miss. If her ship hasn't already been hit, dread pirate Umaima has probability $\dfrac{1}{7}$ of hitting the Captain's ship. If her ship has been hit, dread pirate Umaima will always miss. If the Captain and the pirate each shoot once, and the pirate shoots first, what is the probability that the pirate misses the Captain's ship, but the Captain hits?
Solution: The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened. In this case, event A is the pirate missing the Captain's ship and event B is the Captain hitting the pirate ship. The pirate fires first, so her ship can't be sunk before she fires her cannons. So, the probability of the pirate missing the Captain's ship is $\dfrac{6}{7}$. If the pirate missed the Captain's ship, the Captain has a normal chance to fire back. So, the probability of the Captain hitting the pirate ship given the pirate missing the Captain's ship is $\dfrac{3}{5}$. The probability that the pirate misses the Captain's ship, but the Captain hits is then the probability of the pirate missing the Captain's ship times the probability of the Captain hitting the pirate ship given the pirate missing the Captain's ship. This is $\dfrac{6}{7} \cdot \dfrac{3}{5} = \dfrac{18}{35}$